Random 3-manifolds can be defined in many ways, but the most tractable definition to date is due to Dunfield and Thurston, which involves gluing two handlebodies of genus g by a random self-map of the genus g surface. Even simpler is taking a random surface automorphism T and constructing the mapping torus of T. A tractable way to define a random surface automorphism is to take a nice finitely generated subgroup of the mapping class group and look at random words in the generators of increasing length. Properties that have been studied for random fibered 3-manifolds include: the first betti number is generically 1; the log of torsion grows linearly with word length; volume grows

1. Random 3-manifolds
Igor Rivin (Temple U and Brown U),
AMS Meeting, Baltimore March 29-30

2. Random 3-manifolds
What is a random 3-manifold?
Many definitions, most too hard to deal with

3. Random 3-manifolds
The most tractable (to date) definition is due to
N. Dunfield and W. P. Thurston:
Fix an integer g, take two handle bodies of
genus g, glue them by a random self-map
[whatever that means] of surface of genus g.

4. Random fibered 3-
manifolds
Even simpler: take a (random) surface
automorphism T, construct the mapping torus of
T.

5. Random surface
automorphism?
Again, many ways to define, the most tractable
(to date) is: take a (nice, finitely generated)
subgroup of Mod(g), look at random words in
generators of increasing length.

6. Random surface
automorphisms
What is a “nice subgroup”? Depends on what you
want. An example:
Non-elementary subgroup: contains two non-
commuting pseudo-Anosovs.
Torelli-fat subgroup: Image under the Torelli
homomorphism is Zariski dense in Sp(2g, Z).

7. Random fibered
manifolds
J. Maher and I. Rivin: random automorphism is
pseudo-Anosov, so random fibered manifold is
hyperbolic, by Thurston’s theorem.

8. Random fibered manifold
What can we say?
About: homology, volume, bottom eigenvalue of
the Laplacian, Cheeger constant, rank of the
fundamental group…

9. Random fibered manifold:
homology
The first betti number is generically equal to 1.
If the subgroup is Torelli-fat, log of torsion grows
linearly with length of monodromy; there is also
a central limit theorem for the distribution of the
log torsion.

10. Random fibered 3-
manifold: homology

11. Random fibered 3-
manifolds homology

12. Random fibered 3-
manifolds: volume
Two different ways (both using bi-Lipschitz
models), to see that the volume of random
fibered manifold grows at most and at least
linearly. BUT, the truth is more interesting.

13. Random Fibered 3-
manifolds: volume

14. Random fibered 3-
manifolds: volume

15. Random fibered 3-
manifolds
Rank of fundamental group: 2g+1 (mod Biringer-
Souto)
Cheeger constant: C g/N
injectivity radius 1/log squared N
Bottom eigenvalue between C/N and C/N2.

16. Random 3-manifolds
First betti number is generically zero (already
known to Dunfield-Thurston).
log torsion grows linearly (with central limit and
large deviation thms, which gives exponentially
small probability of nonzero first betti number.

17. Random 3-manifolds
Complexity grows linearly
generically hyperbolic (J. Maher)
Volume grows linearly (as before, can show
linear lower and upper bounds).

18. Random 3-manifolds

19. Random 3-manifolds
Probability of having Galois cover with a fixed
solvable deck group have nontrivial first Betti
number goes to zero (uses
Grunewald/Larsen/Lubotzky/Malestein).

20. Random 3-manifolds
Question: is a random Heegaard splitting
Haken?
By K. Hartshorn, we know that the minimal
genus of an essential surface grows linearly, but
that is neither here nor there.